Abstract

The finite-element approach to the eigenmode analysis of a photonic bandgap cavity by use of an anisotropic perfectly matched layer absorbing boundary is presented. This method rigorously calculates the resonant frequency, the field pattern, and the quality factor of the resonant mode of a finite-sized cavity in free space. The validity of the approach is examined through its application to two-dimensional photonic bandgap cavities. Analyses of numerical error for the resonant frequencies and the quality factor of the cavities demonstrate the accuracy and reliability of our approach, which used nonuniform grids, higher-order elements, and the perfectly matched layer. Far-field patterns of the resonant modes were obtained by simple transformation. Because the perfectly matched layer can represent the real boundary condition well, cavities of any size and shape can be analyzed with the desired accuracy.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  8. E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
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    [CrossRef]
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  32. K. Hayata, M. Eguchi, and M. Koshiba, “Self-consistent finite/infinite element scheme for unbounded guided wave problems,” IEEE Trans. Microwave Theory Tech. MTT-36, 614–616 (1988).
    [CrossRef]
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1997 (5)

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B 56, 14905–14908 (1997).
[CrossRef]

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

S. Hyun, J. Hwang, Y. Lee, and S. Kim, “Computation of resonant modes of open resonators using the FEM and the anisotropic perfectly matched layer boundary condition,” Microwave Opt. Technol. Lett. 16, 352–356 (1997).
[CrossRef]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

1996 (5)

C. C. Cheng and A. Scherer, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4114 (1996).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

A. R. McGurn, “Green’s-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059–7064 (1996).
[CrossRef]

1995 (2)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[CrossRef]

1994 (2)

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1993 (2)

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

1992 (3)

J. B. Pendry and A. Mackinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

R. Lee and A. C. Cangellaris, “A study of discretization error in the finite element approximation of wave solutions,” IEEE Trans. Antennas Propag. 40, 542–549 (1992).
[CrossRef]

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square lattices: square and circular rods,” Phys. Rev. B 46, 4973–4975 (1992).
[CrossRef]

1991 (3)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

1988 (2)

C. R. I. Emson, “Methods for the solution of open-boundary electromagnetic-field problems,” Proc. Inst. Electr. Eng. Part A 135, 151–158 (1988).

K. Hayata, M. Eguchi, and M. Koshiba, “Self-consistent finite/infinite element scheme for unbounded guided wave problems,” IEEE Trans. Microwave Theory Tech. MTT-36, 614–616 (1988).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Alerhand, O. L.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Busch, A.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Cangellaris, A. C.

R. Lee and A. C. Cangellaris, “A study of discretization error in the finite element approximation of wave solutions,” IEEE Trans. Antennas Propag. 40, 542–549 (1992).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[CrossRef]

Cheng, C. C.

C. C. Cheng and A. Scherer, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4114 (1996).
[CrossRef]

Devenyi, A.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[CrossRef]

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Eguchi, M.

K. Hayata, M. Eguchi, and M. Koshiba, “Self-consistent finite/infinite element scheme for unbounded guided wave problems,” IEEE Trans. Microwave Theory Tech. MTT-36, 614–616 (1988).
[CrossRef]

Emson, C. R. I.

C. R. I. Emson, “Methods for the solution of open-boundary electromagnetic-field problems,” Proc. Inst. Electr. Eng. Part A 135, 151–158 (1988).

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[CrossRef]

Felbacq, D.

D. Maystre, G. Tayeb, and D. Felbacq, “Electromagnetic study of photonic band structures and anderson localization,” in Microcavities and Photonic Bandgaps: Physics and Applications, J. Rarity and C. Weisbuch, eds. (Kluwer, Dordrecht, The Netherlands, 1996), pp. 153–163.

Gmitter, T. M.

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Hayata, K.

K. Hayata, M. Eguchi, and M. Koshiba, “Self-consistent finite/infinite element scheme for unbounded guided wave problems,” IEEE Trans. Microwave Theory Tech. MTT-36, 614–616 (1988).
[CrossRef]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Hwang, J.

S. Hyun, J. Hwang, Y. Lee, and S. Kim, “Computation of resonant modes of open resonators using the FEM and the anisotropic perfectly matched layer boundary condition,” Microwave Opt. Technol. Lett. 16, 352–356 (1997).
[CrossRef]

Hyun, S.

S. Hyun, J. Hwang, Y. Lee, and S. Kim, “Computation of resonant modes of open resonators using the FEM and the anisotropic perfectly matched layer boundary condition,” Microwave Opt. Technol. Lett. 16, 352–356 (1997).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley-Interscience, New York, 1993), Chaps. 4, 7.

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[CrossRef]

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S. R.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Kanskar, M.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Kash, K.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Kim, S.

S. Hyun, J. Hwang, Y. Lee, and S. Kim, “Computation of resonant modes of open resonators using the FEM and the anisotropic perfectly matched layer boundary condition,” Microwave Opt. Technol. Lett. 16, 352–356 (1997).
[CrossRef]

Kingsland, D. M.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Koshiba, M.

K. Hayata, M. Eguchi, and M. Koshiba, “Self-consistent finite/infinite element scheme for unbounded guided wave problems,” IEEE Trans. Microwave Theory Tech. MTT-36, 614–616 (1988).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Lee, J.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

R. Lee and A. C. Cangellaris, “A study of discretization error in the finite element approximation of wave solutions,” IEEE Trans. Antennas Propag. 40, 542–549 (1992).
[CrossRef]

Lee, Y.

S. Hyun, J. Hwang, Y. Lee, and S. Kim, “Computation of resonant modes of open resonators using the FEM and the anisotropic perfectly matched layer boundary condition,” Microwave Opt. Technol. Lett. 16, 352–356 (1997).
[CrossRef]

Leung, K. M.

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

Mackenzie, J.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Mackinnon, A.

J. B. Pendry and A. Mackinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

A. A. Maradudin and A. R. McGurn, “Photonic band structures of two-dimensional dielectric media,” in Photonic Band Gap and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 247–268.

Maystre, D.

D. Maystre, G. Tayeb, and D. Felbacq, “Electromagnetic study of photonic band structures and anderson localization,” in Microcavities and Photonic Bandgaps: Physics and Applications, J. Rarity and C. Weisbuch, eds. (Kluwer, Dordrecht, The Netherlands, 1996), pp. 153–163.

McGurn, A. R.

A. R. McGurn, “Green’s-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059–7064 (1996).
[CrossRef]

A. A. Maradudin and A. R. McGurn, “Photonic band structures of two-dimensional dielectric media,” in Photonic Band Gap and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 247–268.

Meade, R. D.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[CrossRef]

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Morin, R.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Ohtaka, K.

K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B 56, 14905–14908 (1997).
[CrossRef]

Pacradoui, V.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Paddon, P.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Pendry, J. B.

J. B. Pendry and A. Mackinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Piché, M.

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square lattices: square and circular rods,” Phys. Rev. B 46, 4973–4975 (1992).
[CrossRef]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Qiu, Y.

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Reddy, J. N.

J. N. Reddy, An Introduction to the Finite Element Method (McGraw-Hill, Singapore, 1993), Chap. 9.

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Sakoda, K.

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B 56, 14905–14908 (1997).
[CrossRef]

Scherer, A.

C. C. Cheng and A. Scherer, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4114 (1996).
[CrossRef]

Shiroma, H.

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

Smith, D. A.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Boston, Mass., 1995), Chaps. 5, 8, 11.

Tayeb, G.

D. Maystre, G. Tayeb, and D. Felbacq, “Electromagnetic study of photonic band structures and anderson localization,” in Microcavities and Photonic Bandgaps: Physics and Applications, J. Rarity and C. Weisbuch, eds. (Kluwer, Dordrecht, The Netherlands, 1996), pp. 153–163.

Tiedje, T.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Ueta, T.

K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B 56, 14905–14908 (1997).
[CrossRef]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square lattices: square and circular rods,” Phys. Rev. B 46, 4973–4975 (1992).
[CrossRef]

Winn, J. N.

Yablonovitch, E.

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Young, J. F.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Zienkiewicz, O. C.

O. C. Zienkiewicz, The Finite Element Method (McGraw-Hill, Singapore, 1989), Vol. 1, Chap. 8.

Appl. Phys. Lett. (1)

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

R. Lee and A. C. Cangellaris, “A study of discretization error in the finite element approximation of wave solutions,” IEEE Trans. Antennas Propag. 40, 542–549 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Hayata, M. Eguchi, and M. Koshiba, “Self-consistent finite/infinite element scheme for unbounded guided wave problems,” IEEE Trans. Microwave Theory Tech. MTT-36, 614–616 (1988).
[CrossRef]

J. Appl. Phys. (1)

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash, “Novel applications of photonic band gap materials: low loss bends and high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Vac. Sci. Technol. B (1)

C. C. Cheng and A. Scherer, “Lithographic band gap tuning in photonic band gap crystals,” J. Vac. Sci. Technol. B 14, 4110–4114 (1996).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

S. Hyun, J. Hwang, Y. Lee, and S. Kim, “Computation of resonant modes of open resonators using the FEM and the anisotropic perfectly matched layer boundary condition,” Microwave Opt. Technol. Lett. 16, 352–356 (1997).
[CrossRef]

Nature (London) (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Phys. Rev. B (10)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square lattices: square and circular rods,” Phys. Rev. B 46, 4973–4975 (1992).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

A. R. McGurn, “Green’s-function theory for row and periodic defect arrays in photonic band structures,” Phys. Rev. B 53, 7059–7064 (1996).
[CrossRef]

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[CrossRef]

K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B 56, 14905–14908 (1997).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

K. M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

Phys. Rev. Lett. (6)

J. B. Pendry and A. Mackinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in a photonic crystal waveguide,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, T. M. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Proc. Inst. Electr. Eng. Part A (1)

C. R. I. Emson, “Methods for the solution of open-boundary electromagnetic-field problems,” Proc. Inst. Electr. Eng. Part A 135, 151–158 (1988).

Other (6)

J. N. Reddy, An Introduction to the Finite Element Method (McGraw-Hill, Singapore, 1993), Chap. 9.

O. C. Zienkiewicz, The Finite Element Method (McGraw-Hill, Singapore, 1989), Vol. 1, Chap. 8.

D. Maystre, G. Tayeb, and D. Felbacq, “Electromagnetic study of photonic band structures and anderson localization,” in Microcavities and Photonic Bandgaps: Physics and Applications, J. Rarity and C. Weisbuch, eds. (Kluwer, Dordrecht, The Netherlands, 1996), pp. 153–163.

J. Jin, The Finite Element Method in Electromagnetics (Wiley-Interscience, New York, 1993), Chaps. 4, 7.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Boston, Mass., 1995), Chaps. 5, 8, 11.

A. A. Maradudin and A. R. McGurn, “Photonic band structures of two-dimensional dielectric media,” in Photonic Band Gap and Localization, C. M. Soukoulis, ed. (Plenum, New York, 1993), pp. 247–268.

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Figures (9)

Fig. 1
Fig. 1

Geometry of a 2-D photonic bandgap cavity surrounded by a PML.

Fig. 2
Fig. 2

PML region where the interface plane is normal to the x direction. A wave incident upon the PML region is transmitted to the region with no reflection.

Fig. 3
Fig. 3

(a) Linear and (b) quadratic serendipity elements.

Fig. 4
Fig. 4

Boundary approximation with linear and quadratic elements. The true boundary of the domain can be represented by the quadratic elements.

Fig. 5
Fig. 5

(a) Resonant mode pattern with quadratic elements for a 5×5 photonic bandgap cavity on a logarithmic scale. Each rod has a refractive index of 3.4 and a radius of rods r=0.2a. A parabolic conductivity profile with R=1×10-6 is used. λ0/δx=19.4, r=0.48λ0, d=0.83λ0. (b) Ez field at y=0. Ez is normalized and phase shifted as explained in the text.

Fig. 6
Fig. 6

(a) Normalized error in the resonant frequency of a 5×5 cavity versus the average nodal density in the linear and quadratic elements. A parabolic absorption function with R=1×10-6 is used for the PML. rλ0/2, d3λ0/4. (b) Normalized error in the quality factor versus the nodal density.

Fig. 7
Fig. 7

Normalized error in the quality factor versus the distance between the structure and the PML for the different thickness of the PML. The number of layers used for the PML is denoted in parentheses. m=2, R=1×10-6.

Fig. 8
Fig. 8

(a) Resonant frequency versus cavity size with the quadratic elements. λ0/δx=19.4, m=2, R=1×10-6, d3λ0/4, rλ0/2. (b) Quality factor versus structure size.

Fig. 9
Fig. 9

Far-field patterns of the resonant modes of 5×5 and 11×11 cavities with quadratic elements. λ0/δx=19.4, m=2, R=1×10-6, d3λ0/4, rλ0/2.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

×E+jω[μr]H=0,
×H-jω[r]E=0,
[r]=xx000yy000zz,[μr]=μxx000μyy000μzz.
E(x, y, z)=E(x, y)exp(-jkzz),
H(x, y, z)=H(x, y)exp(-jkzz),
-x 1μyy Ezx-y 1μxx Ezy-zzω2Ez=0,
-x 1yy Hzx-y 1xx Hzy-μzzω2Hz=0.
F(Ez)=12 ΩαxEzx2+αyEzy2+βzEz2dΩ,
F(Ez)=12 Ω+ΩextαxEzx2+αyEzy2+βzEz2dΩ=12 ΩαxEzx2+αyEzy2+βzEz2dΩ-12 ΓEzαx Ezx xˆ+αy Ezy yˆnˆdΓ,
Fb=-12 ΓEzαx Ezx xˆ+αy Ezy yˆnˆdΓ
xx=μxx=1α-jβ(x),
yy=μyy=zz=μzz=α-jβ(x)
β(x)=-(m+1)ln R2k0d xdm,
E(x, y)=E0 exp[-xωβ(x)cos θi]×exp[-jω(sin θiy+cos θix)],
xx=μxx=zz=μzz=α(x)-jβ(x),
yy=μyy=1α(x)-jβ(x)
Ezh(x, y)=i=1nNie(x, y)Ez,i,
[A]{Ez}-ω2[B]{Ez}={0},
Aije=Ωe1μyy(x, y) Niex Njex+1μxx(x, y) Niey NjeydΩ,
Bije=Ωezz(x, y)NieNjedΩ.
Ezh(x, y)=i=18aixjyk(j+k3,j, k2),
Ez(r)=Ca[G(r|r)n^aEz(r)-Ez(r)n^aG(r|r)]dC,
G(r|r)=j4 H0(2)(k|r-r|).
limk|r-r| Ez(r)=exp(j3π/4)8πkr exp(-jkr)Ce[n^aEz(r)-jkEz(r)n^arˆ]×exp(jkrˆr)dC.
NRCS(θ)=limr |Ez(rθ)|2|Ez(rx)|2=Ca[n^aEz(r)-jkEz(r)nar^θ]exp(jkr^θr)dC2Ca[n^aEz(r)-jkEz(r)naxˆ]exp(jkxˆr)dC2.

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